Continuity

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This editable Main Article is under development and subject to a disclaimer.

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In mathematics, the notion of continuity of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.

## Contents

* 1 Formal definitions of continuity
* 1.1 Function of a real variable
* 1.2 Function on a metric space
* 1.3 Function on a topological space
* 2 Continuous function

## Formal definitions of continuity[edit]

We can develop the definition of continuity from the  δ − ϵ {\displaystyle \delta -\epsilon } formalism which are usually taught in first year calculus courses to general topological spaces.

### Function of a real variable[edit]

The  δ − ϵ {\displaystyle \delta -\epsilon } formalism defines limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at  x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } if (it is defined in a neighborhood of  x 0 {\displaystyle x_{0}} and) for any  ε > 0 {\displaystyle \varepsilon >0} there exist  δ > 0 {\displaystyle \delta >0} such that

| x − x 0 | < δ ⟹ | f ( x ) − f ( x 0 ) | < ε . {\displaystyle |x-x_{0}|<\delta \implies |f(x)-f(x_{0})|<\varepsilon .\,}

Simply stated, the limit

lim x -> x 0 f ( x ) = f ( x 0 ) . {\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0}).}

This definition of continuity extends directly to functions of a complex variable.

### Function on a metric space[edit]

A function f from a metric space ( X , d ) {\displaystyle (X,d)} to another metric space  ( Y , e ) {\displaystyle (Y,e)} is continuous at a point  x 0 ∈ X {\displaystyle x_{0}\in X} if for all  ε > 0 {\displaystyle \varepsilon >0} there exists  δ > 0 {\displaystyle \delta >0} such that

d ( x , x 0 ) < δ ⟹ e ( f ( x ) , f ( x 0 ) ) < ε . {\displaystyle d(x,x_{0})<\delta \implies e(f(x),f(x_{0}))<\varepsilon .\,}

If we let  B d ( x , r ) {\displaystyle B_{d}(x,r)} denote the open ball of radius r round x in X, and similarly  B e ( y , r ) {\displaystyle B_{e}(y,r)} denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back  f ⊣ {\displaystyle f^{\dashv }}

f ⊣ [ B e ( f ( x ) , ε ) ] ⊇ B d ( x , δ ) . {\displaystyle f^{\dashv }[B_{e}(f(x),\varepsilon )]\supseteq B_{d}(x,\delta ).\,}

### Function on a topological space[edit]

A function f from a topological space ( X , O X ) {\displaystyle (X,O_{X})} to another topological space  ( Y , O Y ) {\displaystyle (Y,O_{Y})} , usually written as  f : ( X , O X ) -> ( Y , O Y ) {\displaystyle f:(X,O_{X})\rightarrow (Y,O_{Y})} , is said to be continuous at the point  x ∈ X {\displaystyle x\in X} if for every open set U y ∈ O Y {\displaystyle U_{y}\in O_{Y}} containing the point y=f(x), there exists an open set  U x ∈ O X {\displaystyle U_{x}\in O_{X}} containing x such that  f ( U x ) ⊂ U y {\displaystyle f(U_{x})\subset U_{y}} . Here  f ( U x ) = { f ( x ′ ) ∈ Y ∣ x ′ ∈ U x } {\displaystyle f(U_{x})=\\{f(x')\in Y\mid x'\in U_{x}\\}} . In a variation of this definition, instead of being open sets,  U x {\displaystyle U_{x}} and  U y {\displaystyle U_{y}} can be taken to be, respectively, a neighbourhood of x and a neighbourhood of  y = f ( x ) {\displaystyle y=f(x)} .

## Continuous function[edit]

If the function f is continuous at every point  x ∈ X {\displaystyle x\in X} then it is said to be a continuous function. There is another important equivalent definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis. A function  f : ( X , O X ) -> ( Y , O Y ) {\displaystyle f:(X,O_{X})\rightarrow (Y,O_{Y})} is said to be continuous if for any open set  U ∈ O Y {\displaystyle U\in O_{Y}} (respectively, closed subset of Y ) the set  f − 1 ( U ) = { x ∈ X ∣ f ( x ) ∈ U } {\displaystyle f^{-1}(U)=\\{x\in X\mid f(x)\in U\\}} is an open set in  O x {\displaystyle O_{x}} (respectively, a closed subset of X).